| L(s) = 1 | + 2-s + 0.649·3-s + 4-s + 5-s + 0.649·6-s + 1.95·7-s + 8-s − 2.57·9-s + 10-s − 0.533·11-s + 0.649·12-s + 3.04·13-s + 1.95·14-s + 0.649·15-s + 16-s + 3.79·17-s − 2.57·18-s + 2.82·19-s + 20-s + 1.27·21-s − 0.533·22-s − 6.17·23-s + 0.649·24-s + 25-s + 3.04·26-s − 3.62·27-s + 1.95·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.375·3-s + 0.5·4-s + 0.447·5-s + 0.265·6-s + 0.739·7-s + 0.353·8-s − 0.859·9-s + 0.316·10-s − 0.160·11-s + 0.187·12-s + 0.843·13-s + 0.523·14-s + 0.167·15-s + 0.250·16-s + 0.919·17-s − 0.607·18-s + 0.648·19-s + 0.223·20-s + 0.277·21-s − 0.113·22-s − 1.28·23-s + 0.132·24-s + 0.200·25-s + 0.596·26-s − 0.697·27-s + 0.369·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.562818520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.562818520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 - 0.649T + 3T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 + 0.533T + 11T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 7.15T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 + 8.22T + 47T^{2} \) |
| 53 | \( 1 - 0.727T + 53T^{2} \) |
| 59 | \( 1 + 8.03T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 5.92T + 67T^{2} \) |
| 71 | \( 1 - 6.25T + 71T^{2} \) |
| 73 | \( 1 - 3.20T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.042224847479920864913505500182, −7.53289957660405376380839727427, −6.31758401478032802205731036468, −5.96246342295286380163750022995, −5.22463018641203966364505175429, −4.49239732221329858607750239857, −3.56137247266865779416691205527, −2.88579153132630523289098727002, −2.03972316695745648208206779811, −1.04791967050703562817934540897,
1.04791967050703562817934540897, 2.03972316695745648208206779811, 2.88579153132630523289098727002, 3.56137247266865779416691205527, 4.49239732221329858607750239857, 5.22463018641203966364505175429, 5.96246342295286380163750022995, 6.31758401478032802205731036468, 7.53289957660405376380839727427, 8.042224847479920864913505500182
